From Historical Companion to Hymns Ancient & Modern (1962, p. 168):
NOEL. The first four lines are from a traditional carol melody, received from a friend by Sir Arthur Sullivan, who slightly modified it, added four lines, and printed it in Church Hymns, 1874, of which he was editor. The tune given by Bramley and Steiner in Christmas Carols New & Old, No. 39, appears to be a variant of this tune. Compare also No. 5 in the Appendix to The Oxford Book of Carols. Both this and the previous tune are set to the "Dives and Lazarus" Carol.